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Theorem ax10o 1644
 Description: Show that ax-10o 1645 can be derived from ax-10 1437. An open problem is whether this theorem can be derived from ax-10 1437 and the others when ax-11 1438 is replaced with ax-11o 1745. See theorem ax10 1646 for the rederivation of ax-10 1437 from ax10o 1644. Normally, ax10o 1644 should be used rather than ax-10o 1645, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.)
Assertion
Ref Expression
ax10o (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Proof of Theorem ax10o
StepHypRef Expression
1 ax-10 1437 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
2 ax-11 1438 . . . 4 (𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥𝜑)))
32equcoms 1635 . . 3 (𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥𝜑)))
43sps 1471 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥𝜑)))
5 pm2.27 39 . . 3 (𝑦 = 𝑥 → ((𝑦 = 𝑥𝜑) → 𝜑))
65al2imi 1388 . 2 (∀𝑦 𝑦 = 𝑥 → (∀𝑦(𝑦 = 𝑥𝜑) → ∀𝑦𝜑))
71, 4, 6sylsyld 57 1 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1283 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-5 1377  ax-gen 1379  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464 This theorem depends on definitions:  df-bi 115 This theorem is referenced by:  hbae  1647  dral1  1659
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